Microscopic modelling of ignition and burning for well-arranged energetic crystals in response to drop-weight impact

To probe into impact sensitivity of energetic crystals, a theoretical approach was developed for modelling a single layer of energetic particles between upper striker and below base. Considering the particle plasticity, frictional heating, melting, fracture, and chemical reaction at particle level, effects of loading parameters and sample characteristics on time-to-ignition and burning rate were compared. Finite element numerical simulations were simultaneously performed to provide supporting evidence for thermo-mechanical interactions among energetic particles. Once hot- spots ignition occurred during impact, the macrokinetics of chemical reactions were formulated by hot-spots density, combustion wave velocity and geometric factor. The resulting reaction may or may not develop into a violent event, may be sustained or be extinguished, which can be revealed from the subsequent burn reaction rate.


Introduction
It has long been necessary to postulate the existence of an energy localization process in order to explain the sensitivity of explosives to low-level impact. The drop-weight test is the simplest and most commonly used method of assessing the sensitivity of low-velocity impacted explosives [1,2]. However, drop weight test sensitivity data from various laboratories usually are not consistent with each other. Most sensitivity evaluation tests focus on experimental determinations of 'go/no go' conditions [3,4], with limited efforts toward identifying the underlying ignition micro-mechanisms.
There are many possible hot spot sources at nano-or microscopic scale, including dislocations, voids, twins, inclusions. However, when dropping a hammer onto a granular explosive charge, contacts with other particles or with the solid walls at mesoscopic scale play the key role for temperature fluctuation or deformation localization at engineering scale. Friedman correlated thermal explosion with the impact sensitivities of several explosives [5]. Francis and Cuthbert used finite element analysis of the dropweight test to calculate the temperature in the localized zones near the sample periphery [6]. In the drop-weight system, ignition mechanisms such as adiabatic shear bands, trapped gas spaces, viscous flow, friction, fracture and triboluminescent discharge have ever been discussed [7].
The present model was established by working out kinematic equations for a layer of wellarranged impacted HMX particles, considering frictional heating, micro-impact plasticity and breakage of particles. It is capable of describing the following events: 1) how the drop weight mechanical energy will be transferred to the sample, 2) what kind of localized energy that is focused in local regions upon impact, 3) whether or not the reaction developed from these hot spots into a violent burning.

Finite element model
The finite element simulations were performed based on commercial software code ABAQUS/Explicit. Axisymmetric displacement boundary conditions were imposed on a quarter of the cylindrical striker, base anvil and target explosive charge, as shown in figure 1. All outside surfaces remained to be ambient temperature (300K). Current simulation used simple thermal elastic-plastic constitutive equations for the energetic crystals [8] and elastic responses for striker and base. There existed friction heating, plastic dissipation energy and heat conduction between neighbour particles, and between particles with impacting contact surfaces.    Figure 3 presented the pressure contours for the energetic grains subjected to 20cm drop-weight impact. Pressure localization can be observed at the contact surfaces for each particle at 250µs, as well as in the centre of the charge with large-scale linkages were formed till 500µs. The degree of pressure concentration in the centre was more serious at time 700µs than at 500µs. There appeared tensile pressure regions along the boundary of explosive charge with the development of impact process, as shown in figure 5(b) and (c). a)250µs b) 500µs c) 700µs Figure 3 Simulated pressure distributions over the energetic grains at different moment subjected to 7.36 kg drop-weight and 20 cm drop height. Figure 4 showed the heat flux leaving each particle surface per unit area through three different paths: grains interface; upper impact surface and below anvil surface. Once the impact began, the upper dropped surface took away heat from particle's surface (at about 0.04ms). Subsequently, with the stress wave propagated, heat flux leaving the grains from below surface needed to be taken into account (at about 0.11ms). When grains deformed to contact with each other, heat flux leaving grain's interface performed a key role (at about 0.3ms). Heat flux in figure 4 implied that heat conduction can be formulated based on one-dimensional system at early stage and then should be transformed to cylindrical-coordinate system for later stage.

Mechanics analysis
Based on numerical simulated results, the explosive charge composing of well-arranged energetic crystals without overlaps impacted by the upper dropping striker was modelled. Figure5 (a) explained the approach for modelling a single layer of separate crystals between striker and base. Figure5(b) provides an illustrative graph of contact mechanics for an individual particle prior to fragmentation. To simplify the calculations, energetic crystals were assumed be equal-sized spherical particles with initial diameter d 0 (mm) and spacing of w 0 (mm). The explosive charge occupied an initial circular area with diameter d s0 (mm). The motion of the dropped striker during impact was determined from its momentum balance, where x u denoted the falling distance of the striker, g is the acceleration of gravity, and F c (t) is the force applied to the explosive charge by the weight. A simple relationship between the energy absorbed by sample (W sample ) and the input kinetic energy (W inp ) was used, Dependence of coefficient f eff ⋅ on initial drop height H 0 can be obtained based on the numerical results. More detailed formulas can refer to the literature [9,10].
(a) (b) Figure 5 Schematic graphs explaining the drop-weight impact process with (a) global drop-weight model, (b) contact deformation of individual particle.

Thermal analysis
In the course of an impact event, heat generation by friction, irreversible plastic work, and chemical reaction heat, as well as heat conduction through the contact surface, determined the flash temperature rise at those hot spots. Prior to ignition, each particle was regarded as the separated and linked heat sources at different stages as shown in figure 6(a).
(a) (b) Figure 6 Thermal model for analyzing ignition and hot spots growth to burning a) separated heat source before ignition; b) linked heat source when severe deformation occurs.
where α, M, D c were the constant, density of ignited hot spots, the propagation velocity of the combustion wave, respectively. The geometric factor F was introduced to take into account the change in combustion topology. Figure 7(a) presented times-to-ignition vs. frictional coefficients between HMX and impact surface subjected to different drop-height impacts, keeping other conditions the same. It showed that time-toignition decreased with the increase of frictional coefficients. Each curve provided a threshold value of frictional coefficients, below which ignition never occurred. A careful examination demonstrated that ignition occurred later with freshly polished surfaces than rougher surfaces [13]. In figure 7(b), a series of calculations were conducted by altering Young′s modulus of anvil materials. Time-to-ignition decreased with the increase of Young′s modulus. Calculated results showed that harder anvil material leads to shorter loading time and promotes the ignition process.   Figure 8 provided the effects of drop heights, drop weights and porosity on the macroscopic burn reaction rate evolutions. The burning reaction rate showed that the heavier the drop weight was, the more intense the burn chemical reaction was. Different loose degree distribution of grains influenced both macroscopic burn reaction beginning time and its violence.

Conclusions
It has long been necessary to postulate the existence of an energy localization process in order to explain the sensitivity of explosives subjected to low-level impact. Numerical simulated results provided the heat flux and mechanical interactions among particles, and particles and contact surface. The results supported our assumptions proposed for subsequent theoretical model. Based on assumptions including given absorbed energy, equal-sized spherical particles, one-dimensional wave propagation, a micromechanics theoretical approach was developed to calculate hot spots formation and growth to burning for a single layer of impacted energetic particles. Threshold frictional coefficients and Young′s modulus of anvil material for ignition to occur can be calculated for the HMX granular explosives. The calculated results implied that simple ordering of sensitivity for explosive materials based on drop weight impact experiments is very difficult.